a(n) = Sum_{k=0..floor(n/4)} 2^(n-4*k) * binomial(2*n-6*k+1,2*k+1).

Terms

a(0) =1a(1) =6a(2) =20a(3) =56a(4) =145a(5) =372a(6) =972a(7) =2592a(8) =6993a(9) =18922a(10) =51128a(11) =137840a(12) =371057a(13) =998312a(14) =2685992a(15) =7228320a(16) =19455793a(17) =52371822a(18) =140978220a(19) =379488360a(20) =1021494369a(21) =2749592540a(22) =7401150420a(23) =19921892800a(24) =53624455905a(25) =144343043154a(26) =388534041776